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6) a) Let the function f been given by S1, f(x) = {0, when 0 < x < 1 when x = 1 Show that for each e > 0 there is a partition P, of [0,1] such that for the lower Riemann sum L(f, P.) applies that L(f, P.) > 1 – €.

6) a) Let the function f been given by S1, f(x) = {0, when 0 < x < 1 when x = 1 Show that for each e > 0 there is a partition P, of [0,1] such that for the lower Riemann sum L(f, P.) applies that L(f, P.) > 1 – €.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 64E
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6) a) Let the function f been given by (1, when 0 < x < 1 f(x) = (o, when x = 1 Show that for each e > 0 there is a partition P, of [0,1] such that for the lower Riemann sum L(f,P.) applies that L(f, P.) > 1 - €. b) Use (a) to show that f is integrable and that f(x)dx = 1 c) Let g be a decreasing continuous function on [0,1], assuming the values g(0) = 1 and g(1) = 0. Show that there is a partition P of [0,1] such that the following inequalities apply to the lower and the upper Riemann sum. 0 < L(g, P)< U(g,P) < 1 d) Use c) to show the inequalities 0 < g(x)dx < 1 (You can use that a continuous function on [0,1] is integrable)
Transcribed Image Text:6) a) Let the function f been given by (1, when 0 < x < 1 f(x) = (o, when x = 1 Show that for each e > 0 there is a partition P, of [0,1] such that for the lower Riemann sum L(f,P.) applies that L(f, P.) > 1 - €. b) Use (a) to show that f is integrable and that f(x)dx = 1 c) Let g be a decreasing continuous function on [0,1], assuming the values g(0) = 1 and g(1) = 0. Show that there is a partition P of [0,1] such that the following inequalities apply to the lower and the upper Riemann sum. 0 < L(g, P)< U(g,P) < 1 d) Use c) to show the inequalities 0 < g(x)dx < 1 (You can use that a continuous function on [0,1] is integrable)
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